Lifts of Brauer characters in characteristic two, II

TL;DR

Develops a streamlined Navarro-vertex framework to count lifts of Brauer characters in characteristic two. Replacing twisted/$*$-vertices with Navarro vertices, it proves a general bound $|L_\varphi(Q,\delta)| \le |\mathbf{N}_G(Q):\mathbf{N}_G(Q,\delta)|$ for $G$ a $\pi$-separable group, yielding $|\tilde{L}_\varphi| \le |Q:Q'|$ and thus recovering Cossey-type bounds in the 2-modular setting. The method works without restriction on $\pi$ and relies on normal nuclei, Clifford theory, and inertial groups to count lifts via Navarro vertices. Applications include consequences for groups of order $p^a q^b$ and related counting results.

Abstract

In 2007, J. P. Cossey conjectured that if $G$ is a finite $p$-solvable group and $\varphi$ is an irreducible Brauer character of $G$ with vertex $Q$, then the number of lifts of $\varphi$ is at most $|Q:Q'|$. In this paper we revisited Cossey's conjecture for $p=2$ from the perspective of Navarro vertices and obtained a new way to count the number of lifts of $\varphi$. Some applications were given.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3